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Question
Let `A= [[1,-1,0],[2,1,3],[1,2,1]]` And `B=[[1,2,3],[2,1,3],[0,1,1]]` Find `A^T,B^T` and verify that (2A)T = 2AT.
Solution
\[Given: A = \begin{bmatrix}1 & - 1 & 0 \\ 2 & 1 & 3 \\ 1 & 2 & 1\end{bmatrix} \text{and B} = \begin{bmatrix}1 & 2 & 3 \\ 2 & 1 & 3 \\ 0 & 1 & 1\end{bmatrix}\] \[2A = 2\begin{bmatrix}1 & - 1 & 0 \\ 2 & 1 & 3 \\ 1 & 2 & 1\end{bmatrix} \]
\[ \Rightarrow 2A = \begin{bmatrix}2 & - 2 & 0 \\ 4 & 2 & 6 \\ 2 & 4 & 2\end{bmatrix}\]
\[ \Rightarrow \left( 2A \right)^T = \begin{bmatrix}2 & 4 & 2 \\ - 2 & 2 & 4 \\ 0 & 6 & 2\end{bmatrix} . . . \left( 1 \right)\]
\[Now, \]
\[2 A^T = 2\begin{bmatrix}1 & 2 & 1 \\ - 1 & 1 & 2 \\ 0 & 3 & 1\end{bmatrix}\]
\[ \Rightarrow 2 A^T = \begin{bmatrix}2 & 4 & 2 \\ - 2 & 2 & 4 \\ 0 & 6 & 2\end{bmatrix} . . . \left( 2 \right)\]
\[ \Rightarrow \left( 2A \right)^T = 2 A^T \left[ \text{From eqs} . (1) and (2) \right]\]
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